The equation of the circle passing through $(1,0)$ and $(0,1)$ and having the smallest possible radius is

Let the equation of the required circle be $x^2+y^2+2 g x+2 f y+c=0...(i)$ This passes through $A(1,0)$ and $B(0,1)$, therefore, $1+2 g+c=0$ and $1+2 f+c=0$ $\Rightarrow \quad g=-\left(\frac{c+1}{2}\right)$ and $f=-\left(\frac{c+1}{2}\right)$ Let $r$ be the radius of circle (i), then $r=\sqrt{g^2+f^2-c}$ $\Rightarrow \quad r=\sqrt{\left(\frac{c+1}{2}\right)^2+\left(\frac{c+1}{2}\right)^2-c}$ $\Rightarrow \quad r=\sqrt{\frac{c^2+1}{2}}$ $\Rightarrow \quad r^2=\frac{1}{2}\left(c^2+1\right)$ Clearly, $r$ is minimum when $c=0$ and the minimum value of $r$ is $\frac{1}{\sqrt{2}}$. For $c=0$, we have $g=-\frac{1}{2} \text { and } f=-\frac{1}{2}$ On substituting the values of $g, f$ and $c$ in Eq. (i), we get, $x^2+y^2-x-y=0$ This is equation of the required circle.